Question: Simplify and expand the following expression: $ \dfrac{p - 1}{4p + 2}+\dfrac{p + 8}{p - 7} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4p + 2)(p - 7)$ Multiply the first term by $\dfrac{p - 7}{p - 7}$ $ \begin{align*} \dfrac{p - 1}{4p + 2} \times \dfrac{p - 7}{p - 7} & = \dfrac{(p - 1)(p - 7)}{(4p + 2)(p - 7)} \\ & = \dfrac{p^2 - 8p + 7}{(4p + 2)(p - 7)}\end{align*} $ Multiply the second term by $\dfrac{4p + 2}{4p + 2}$ $ \begin{align*} \dfrac{p + 8}{p - 7} \times \dfrac{4p + 2}{4p + 2} & = \dfrac{(p + 8)(4p + 2)}{(p - 7)(4p + 2)} \\ & = \dfrac{4p^2 + 34p + 16}{(p - 7)(4p + 2)}\end{align*} $ Now we have: $ = \dfrac{p^2 - 8p + 7}{(4p + 2)(p - 7)} + \dfrac{4p^2 + 34p + 16}{(p - 7)(4p + 2)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{p^2 - 8p + 7 + 4p^2 + 34p + 16}{(4p + 2)(p - 7)} $ $ = \dfrac{5p^2 + 26p + 23}{(4p + 2)(p - 7)}$ Expand the denominator: $ = \dfrac{5p^2 + 26p + 23}{4p^2 - 26p - 14}$